The Power Function | 幂函数

📚 The Power Function | 幂函数

The power function is one of the most fundamental building blocks in A-Level mathematics. It appears everywhere, from simple proportional relationships to complex physical models. In this revision guide, we will explore its definition, graphs, properties, and calculus, ensuring you have a complete understanding for your Edexcel exams.

幂函数是 A-Level 数学中最基础的构建模块之一，几乎无处不在——从简单的正比例关系到复杂的物理模型。在本复习指南中，我们将深入探讨其定义、图像、性质和微积分，帮助你在 Edexcel 考试中全面掌握这一考点。

1. Definition and General Form | 定义与一般形式

A power function is a function of the form f(x) = xⁿ, where n is a real constant. The variable x is called the base and n is called the power, exponent, or index. In the Edexcel specification, you will encounter power functions with integer, rational, and irrational exponents.

幂函数是形如 f(x) = xⁿ 的函数，其中 n 为实常数。变量 x 称为底数，n 称为幂、指数或次数。在 Edexcel 考纲中，你会遇到指数为整数、有理数甚至无理数的幂函数。

For example:

f(x) = x² is a quadratic power function (n = 2).
f(x) = x⁻¹ is the reciprocal function (n = -1).
f(x) = x^{1/2} is the square root function (n = ½).

f(x) = x² 是二次幂函数（n = 2）。
f(x) = x⁻¹ 是反比例函数（n = -1）。
f(x) = x^{1/2} 是平方根函数（n = ½）。

2. Domain and Range | 定义域与值域

The domain and range of a power function depend heavily on the value of n. Here is a quick reference:

幂函数的定义域与值域高度依赖于指数 n 的取值。下面是一个快速参考：

n	Domain	Range
Positive integer	All real numbers	All real numbers (if n odd); y ≥ 0 (if n even)
Negative integer	x ≠ 0	y ≠ 0
Fraction 1/m (m even)	x ≥ 0	y ≥ 0
Fraction 1/m (m odd)	All real numbers	All real numbers

For example, f(x) = x^{1/2} is only defined for x ≥ 0 if we restrict to real outputs. Meanwhile, f(x) = x^{-2} has a domain excluding x = 0, since division by zero is not allowed.

例如，f(x) = x^{1/2} 在实数范围内仅对 x ≥ 0 有定义。而 f(x) = x^{-2} 的定义域必须排除 x = 0，因为除数不能为零。

3. Graphs of Power Functions | 幂函数的图像

The shape of the graph y = xⁿ changes dramatically depending on n. You should be able to sketch these without a calculator.

y = xⁿ 的图像形状随 n 的变化而大不相同。你应该能够在不使用计算器的情况下画出这些草图。

If n > 0: the graph passes through (0,0) and (1,1). For 0 < n < 1, it rises steeply at first then flattens; for n > 1, it rises slowly then becomes steep.
If n < 0: the graph passes through (1,1) and has asymptotes at x = 0 and y = 0.

若 n > 0：图像经过 (0,0) 和 (1,1)。当 0 < n < 1 时，先陡峭上升后趋于平缓；当 n > 1 时，先缓慢上升后变得陡峭。
若 n < 0：图像经过 (1,1)，且以 x = 0 和 y = 0 为渐近线。

Key points: (0,0), (1,1), and (-1, (-1)ⁿ) when valid.

关键点：(0,0)、(1,1)，以及当有意义时的 (-1, (-1)ⁿ)。

4. Special Cases: Integer Powers | 特殊情形：整数幂

When n is a positive integer, xⁿ is a polynomial. The degree n determines the number of turns (at most n-1) and the end behaviour.

当 n 为正整数时，xⁿ 是一个多项式。次数 n 决定了图像的转折数（至多 n-1 次）以及两端的行为。

Even n (e.g. n = 2, 4): The graph is symmetric about the y-axis (even function). As x → ±∞, y → +∞.

偶数 n（例如 n = 2, 4）：图像关于 y 轴对称（偶函数）。当 x → ±∞ 时，y → +∞。

Odd n (e.g. n = 3, 5): The graph has rotational symmetry about the origin (odd function). As x → +∞, y → +∞; as x → -∞, y → -∞.

奇数 n（例如 n = 3, 5）：图像关于原点旋转对称（奇函数）。当 x → +∞ 时，y → +∞；当 x → -∞ 时，y → -∞。

5. Special Cases: Rational and Negative Powers | 特殊情形：有理数与负指数

Rational powers combine roots and integer powers. For example, x^{3/2} = √(x³) = (√x)³. Negative powers indicate reciprocals: x⁻ⁿ = 1/(xⁿ).

有理数幂结合了根式与整数幂。例如，x^{3/2} = √(x³) = (√x)³。负指数表示倒数：x⁻ⁿ = 1/(xⁿ)。

When n = -1, the graph is a rectangular hyperbola with two branches, one in the first quadrant and one in the third quadrant. Its equations are xy = 1 and it has both axes as asymptotes.

当 n = -1 时，图像是双曲线，包含第一象限和第三象限的两支。其方程为 xy = 1，两条坐标轴都是渐近线。

For n = 1/2, the graph is the familiar square root curve, starting at the origin and increasing but with a decreasing gradient.

当 n = 1/2 时，图像是熟悉的平方根曲线，从原点出发递增，但梯度逐渐减小。

6. Properties: Symmetry and Monotonicity | 性质：对称性与单调性

We can classify power functions as even or odd based on their symmetry:

我们可以根据对称性将幂函数分为偶函数或奇函数：

Even function: f(-x) = f(x) for all x in the domain, e.g. x², x⁴, x^{-2}.
Odd function: f(-x) = -f(x) for all x in the domain, e.g. x³, x⁵, x^{-1}, x^{-3}.

偶函数：对定义域内所有 x 满足 f(-x) = f(x)，例如 x²、x⁴、x^{-2}。
奇函数：对定义域内所有 x 满足 f(-x) = -f(x)，例如 x³、x⁵、x^{-1}、x^{-3}。

Monotonicity refers to whether the function is increasing or decreasing. On x > 0, xⁿ is strictly increasing for n > 0 and strictly decreasing for n < 0. On x < 0, the behaviour depends on the sign and parity of n.

单调性指的是函数是递增还是递减。在 x > 0 上，当 n > 0 时 xⁿ 严格递增，当 n < 0 时严格递减。在 x < 0 上，行为取决于 n 的符号与奇偶性。

7. Differentiation of Power Functions | 幂函数的微分

The power rule is a cornerstone of calculus. For any constant n,

幂法则是微积分的基石。对于任意常数 n，

d/dx (xⁿ) = n xⁿ⁻¹

This rule works for all real values of n, provided xⁿ and xⁿ⁻¹ are defined. For example, if y = x⁵, then dy/dx = 5x⁴. If y = x^{-3}, dy/dx = -3x^{-4}. If y = x^{1/2}, dy/dx = ½ x^{-1/2} = 1/(2√x).

此规则适用于所有实数 n，前提是 xⁿ 和 xⁿ⁻¹ 有意义。例如，若 y = x⁵，则 dy/dx = 5x⁴。若 y = x^{-3}，则 dy/dx = -3x^{-4}。若 y = x^{1/2}，则 dy/dx = ½ x^{-1/2} = 1/(2√x)。

Remember: constants multiple stay outside the derivative. For f(x) = 3x², f'(x) = 3 × 2x = 6x.

请记住：常数倍数在求导时保持在外。对于 f(x) = 3x²，f'(x) = 3 × 2x = 6x。

8. Integration of Power Functions | 幂函数的积分

Integration reverses differentiation. For n ≠ -1,

积分是微分的逆运算。当 n ≠ -1 时，

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C

The constant C is the constant of integration. For example, ∫ x³ dx = x⁴/4 + C. For n = -1, the rule fails because we would divide by zero; instead, ∫ x⁻¹ dx = ln|x| + C.

其中 C 是积分常数。例如，∫ x³ dx = x⁴/4 + C。当 n = -1 时，该法则失效，因为我们不能除以零；此时 ∫ x⁻¹ dx = ln|x| + C。

This special case is essential for Edexcel: do not confuse the integral of 1/x with the power rule.

这个特殊情形在 Edexcel 考试中至关重要：不要把 1/x 的积分与幂法则混淆。

9. Solving Equations with Power Functions | 幂函数方程的求解

Equations involving power functions can often be solved by raising both sides to a reciprocal power, or by using substitution when the equation is quadratic in form.

涉及幂函数的方程通常可以通过两边同时取倒数次幂来求解，或者当方程呈现二次形式时使用换元法。

Example: Solve x^{2/3} = 4. Raise both sides to the power 3/2:

例：解方程 x^{2/3} = 4。两边同时取 3/2 次幂：

x = 4^{3/2} = (√4)³ = 2³ = 8

Be careful with even roots: x^{2/3} = 4 also has the solution x = -8? Actually no, because x^{2/3} = (∛x)², and if x = -8, (∛(-8))² = (-2)² = 4, so x = -8 is also a solution when the cube root is defined over real numbers. However, if we interpret x^{2/3} as (x²)^{1/3}, then x = -8 works too. Always check the domain conventions.

注意偶数根的情形：x^{2/3} = 4 是否还有解 x = -8？实际上有的，因为 x^{2/3} = (∛x)²，如果 x = -8，(∛(-8))² = (-2)² = 4，所以 x = -8 也是解（当立方根在实数范围内定义时）。然而，如果我们将 x^{2/3} 理解为 (x²)^{1/3}，则 x = -8 同样成立。务必检查定义域约定。

10. Applications in Modelling | 建模中的应用

Power functions are used to model many real-world relationships, such as:

幂函数被用于模拟许多现实世界的关系，例如：

Physics: kinetic energy E = ½mv² (a power function of v).
Biology: body mass vs. metabolic rate, often modelled as P = kM³/⁴.
Economics: diminishing returns can be modelled by y = axⁿ with 0 < n < 1.

物理：动能 E = ½mv²（关于 v 的幂函数）。
生物：体重与代谢率的关系常建模为 P = kM³/⁴。
经济：边际收益递减可以用 y = axⁿ（0 < n < 1）来建模。

In exam questions, you may need to linearise data: taking logs of both sides of y = axⁿ gives ln y = ln a + n ln x, which is a straight line when plotted as ln y against ln x.

在考试题目中，你可能需要对数据线性化：对 y = axⁿ 两边取对数得到 ln y = ln a + n ln x，如果绘制 ln y 对 ln x 的图像，将得到一条直线。

11. Common Mistakes and Tips | 常见错误与技巧

Here are typical pitfalls that Edexcel students encounter:

以下是 Edexcel 学生常遇到的典型陷阱：

Forgetting that x⁰ = 1 for all x ≠ 0.
Applying the power rule to 1/x as if it were a power function with n = -1 when integrating, and writing x⁰/0 instead of ln|x|.
Assuming that x^{1/2} is defined for negative x when using real numbers.
Dropping the absolute value in ∫ x⁻¹ dx = ln|x| + C when x can be negative.

忘记 x⁰ = 1（对所有 x ≠ 0）。
积分时将 1/x 当作 n = -1 的幂函数直接套用幂法则，写成 x⁰/0，而应该写 ln|x|。
在实数范围内假设 x^{1/2} 对负 x 有定义。
当 x 可能为负时，在 ∫ x⁻¹ dx = ln|x| + C 中丢掉了绝对值。

Always sketch a quick graph to verify your answers, especially for transformations such as y = (x – 2)³ or y = -x².

养成快速画图检验答案的习惯，尤其对于 y = (x – 2)³ 或 y = -x² 这样的变换。

12. Summary | 总结

You should now be comfortable with the power function f(x) = xⁿ, including its domain, range, graphs, symmetry, and calculus. Remember the power rule for differentiation, the exclusive exception of n = -1 for integration, and the use of logarithms to linearise power models. With practice, these questions become routine marks in your exam.

现在你应该对幂函数 f(x) = xⁿ 的定义域、值域、图像、对称性以及微积分有了清晰的掌握。牢记微分中的幂法则、积分中 n = -1 的特殊例外，以及利用对数将幂函数模型线性化的方法。通过练习，这类题目将成为你考试中的常规得分点。

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The Power Function | 幂函数